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Second-order variational analysis in second-order cone programming. (English) Zbl 1440.90079
Summary: The paper conducts a second-order variational analysis for an important class of nonpolyhedral conic programs generated by the so-called second-order/Lorentz/ice-cream cone \({\mathcal{Q}}\). From one hand, we prove that the indicator function of \({\mathcal{Q}}\) is always twice epi-differentiable and apply this result to characterizing the uniqueness of Lagrange multipliers together with an error bound estimate in the general second-order cone programming setting involving twice differentiable data. On the other hand, we precisely calculate the graphical derivative of the normal cone mapping to \({\mathcal{Q}}\) under the metric subregularity constraint qualification and then give an application of the latter result to a complete characterization of isolated calmness for perturbed variational systems associated with second-order cone programs. The obtained results seem to be the first in the literature in these directions for nonpolyhedral problems without imposing any nondegeneracy assumptions.

MSC:
90C31 Sensitivity, stability, parametric optimization
49J52 Nonsmooth analysis
49J53 Set-valued and variational analysis
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[1] Alizadeh, F.; Goldfarb, D., Second-order cone programming, Math. Program., 95, 3-51 (2003) · Zbl 1153.90522
[2] Bauschke, Hh; Borwein, Jm; Li, W., Strong conical hull intersection property, bounded linear regularity, Jameson’s property (G), and error bounds in convex optimization, Math. Program., 86, 135-160 (1999) · Zbl 0998.90088
[3] Bonnans, Jf; Ramírez, Hc, Perturbation analysis of second-order cone programmming problems, Math. Program., 104, 205-227 (2005) · Zbl 1124.90039
[4] Bonnans, Jf; Shapiro, A., Perturbation Analysis of Optimization Problems (2000), New York: Spinger, New York
[5] Chieu, Nh; Hien, Lv, Computation of graphical derivative for a class of normal cone mappings under a very weak condition, SIAM J. Optim., 27, 190-204 (2017) · Zbl 1357.49072
[6] Ding, C.; Sun, D.; Zhang, L., Characterization of the robust isolated calmness for a class of conic programming problems, SIAM J. Optim., 27, 67-90 (2017) · Zbl 1357.49100
[7] Do, Cn, Generalized second-order derivatives of convex functions in reflexive Banach spaces, Trans. Am. Math. Soc., 334, 281-301 (1992) · Zbl 0767.49008
[8] Dontchev, Al; Rockafellar, Rt, Implicit Functions and Solution Mappings: A View from Variational Analysis (2014), New York: Springer, New York
[9] Gfrerer, H.; Mordukhovich, Bs, Complete characterizations of tilt stability in nonlinear programming under weakest qualification conditions, SIAM J. Optim., 25, 2081-2119 (2015) · Zbl 1357.49074
[10] Gfrerer, H.; Mordukhovich, Bs, Robinson stability of parametric constraint systems via variational analysis, SIAM J. Optim., 27, 438-465 (2017) · Zbl 1368.49014
[11] Gfrerer, H.; Outrata, Jv, On computation of generalized derivatives of the normal-cone mapping and their applications, Math. Oper. Res., 41, 1535-1556 (2016) · Zbl 1352.49018
[12] Gfrerer, H.; Outrata, Jv, On the Aubin property of a class of parameterized variational systems, Math. Methods Oper. Res., 86, 443-467 (2017) · Zbl 1385.49007
[13] Gfrerer, H.; Ye, Jj, New constraint qualifications for mathematical programs with equilibrium constraints via variational analysis, SIAM J. Optim., 27, 842-865 (2017) · Zbl 1368.49015
[14] Henrion, R.; Jourani, A.; Outrata, Jv, On the calmness of a class of multifunctions, SIAM J. Optim., 13, 603-618 (2002) · Zbl 1028.49018
[15] Henrion, R.; Outrata, Jv, Calmness of constraint systems with applications, Math. Program, 104, 437-464 (2005) · Zbl 1093.90058
[16] Ioffe, Ad; Outrata, Jv, On metric and calmness qualification conditions in subdifferential calculus, Set-Valued Anal., 16, 199-227 (2008) · Zbl 1156.49013
[17] Izmailov, Af; Solodov, Mv, Newton-Type Methods for Optimization and Variational Problems (2014), New York: Springer, New York
[18] Levy, Ab, Sensitivity of solutions to variational inequalities on Banach spaces, SIAM J. Control Optim., 38, 50-60 (1999) · Zbl 0951.49031
[19] Mordukhovich, Bs, Variational Analysis and Generalized Differentiation, I: Basic Theory; II: Applications (2006), Berlin: Springer, Berlin
[20] Mordukhovich, Bs; Outrata, Jv; Ramírez, Hc, Second-order variational analysis in conic programming with applications to optimality and stability, SIAM J. Optim., 25, 76-101 (2015) · Zbl 1356.49021
[21] Mordukhovich, Bs; Outrata, Jv; Ramírez, Hc, Graphical derivatives and stability analysis for parameterized equilibria with conic constraints, Set-Valued Var. Anal., 23, 687-704 (2015) · Zbl 1330.49013
[22] Mordukhovich, Bs; Outrata, Jv; Sarabi, Me, Full stability of locally optimal solution in second-order cone programming, SIAM J. Optim., 24, 1581-1613 (2014) · Zbl 1311.49062
[23] Mordukhovich, Bs; Sarabi, Me, Critical multipliers in variational systems via second-order generalized differentiation, Math. Program., 169, 605-648 (2018) · Zbl 1407.90314
[24] Outrata, Jv; Ramírez, Hc, On the Aubin property of critical points to perturbed second-order cone programs, SIAM J. Optim., 21, 798-823 (2011) · Zbl 1247.90256
[25] Outrata, Jv; Sun, D., On the coderivative of the projection operator onto the second-order cone, Set-Valued Anal., 16, 999-1014 (2008) · Zbl 1161.49011
[26] Robinson, Sm, Some continuity properties of polyhedral multifunctions, Math. Program. Stud., 14, 206-214 (1981) · Zbl 0449.90090
[27] Rockafellar, Rt, Convex Analysis (1970), Princeton: Princeton University Press, Princeton
[28] Rockafellar, Rt, Proto-differentiability of set-valued mappings and its applications in optimization, Ann. Inst. H. Poincaré Anal. Non Linéaire, 6, 448-482 (1987)
[29] Rockafellar, Rt, First- and second-order epi-differentiability in nonlinear programming, Trans. Am. Math. Soc., 307, 75-108 (1988) · Zbl 0655.49010
[30] Rockafellar, Rt; Wets, Rj-B, Variational Analysis (1998), Berlin: Springer, Berlin
[31] Ruszczynski, Ap, Nonlinear Optimization (2006), Priceton: Princeton University Press, Priceton
[32] Shapiro, A., Nemirovski, A.S.: Duality of Linear Conic Problems. School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA. http://www.optimization-online.org/DB_FILE/2003/12/793.pdf (2003)
[33] Wachsmuth, G., Strong stationarity for optimization problems with complementarity constraints in absence of polyhedricity, Set-Valued Anal., 25, 133-175 (2017) · Zbl 1368.49018
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